Straight Lines
Intersection and geometric constraints
Grade 11
Question:
<p>Given a triangle whose vertices are at <span class="math inline">\((0, 0)\)</span>, <span class="math inline">\((4, 4)\)</span> and <span class="math inline">\((10, 0)\)</span>. A square is drawn in it such that its base is on the x-axis and its two corners are on the 2 sides of the triangle. The area of the square is equal to:</p>
<p>(a) <span class="math inline">\(\frac{400}{49}\)</span></p>
<p>(b) <span class="math inline">\(\frac{400}{25}\)</span></p>
<p>(c) <span class="math inline">\(\frac{625}{16}\)</span></p>
<p>(d) <span class="math inline">\(\frac{625}{49}\)</span></p>
Step-by-Step Solution
Key Concept: A square inscribed in a triangle with its base on the x-axis has its top two vertices lying on the two sides of the triangle. We use the equations of these sides and the constraint that the figure is a square to find its side length.
<p><strong>Step 1: Identify the triangle vertices and sides.</strong></p><p>Vertices: A = (0, 0), B = (4, 4), C = (10, 0).</p><p>The base AC lies on the x-axis from (0, 0) to (10, 0).</p><p>Side AB goes from (0, 0) to (4, 4): equation is y = x.</p><p>Side BC goes from (4, 4) to (10, 0): slope = (0 - 4)/(10 - 4) = -4/6 = -2/3, so y - 0 = -2/3(x - 10), giving y = -2/3(x - 10) = -2x/3 + 20/3.</p><p><strong>Step 2: Set up the square.</strong></p><p>Let the square have side length s. Its base lies on the x-axis with vertices at (p, 0) and (p + s, 0) where p > 0.</p><p>Its top two vertices are at (p, s) and (p + s, s).</p><p><strong>Step 3: Apply constraints from triangle sides.</strong></p><p>The left top vertex (p, s) must lie on side AB (equation y = x):</p><p>s = p ... (i)</p><p>The right top vertex (p + s, s) must lie on side BC (equation y = -2x/3 + 20/3):</p><p>s = -2(p + s)/3 + 20/3</p><p>3s = -2(p + s) + 20</p><p>3s = -2p - 2s + 20</p><p>5s = -2p + 20 ... (ii)</p><p><strong>Step 4: Solve the system.</strong></p><p>From (i): p = s</p><p>Substitute into (ii):</p><p>5s = -2s + 20</p><p>7s = 20</p><p>s = 20/7</p><p><strong>Step 5: Calculate the area.</strong></p><p>Area = s² = (20/7)² = 400/49</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A